Transport coefficients including electron-phonon scattering - Revision history

Csheldon at 11:10, 22 January 2026

2026-01-22T11:10:04Z

← Older revision		Revision as of 11:10, 22 January 2026
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	{{Available\|6.5.0}}		{{Available\|6.5.0}}
	{{NB\|important\|This feature requires [[:Category:HDF5_support \| HDF5 support]].}}		{{NB\|important\|This feature requires [[:Category:HDF5_support \| HDF5 support]].}}
			{{NB\|warning\|There was a [[Known_issues#KnownIssue54 \| known issue]] for electron-phonon calculations using {{TAG\|ISPIN\|2}} for VASP 6.5.0 and 6.5.1 that was fixed in VASP 6.6.0.}}

	The most basic {{FILE\|INCAR}} to compute the transport coefficients is		The most basic {{FILE\|INCAR}} to compute the transport coefficients is

Csheldon at 09:45, 4 November 2025

2025-11-04T09:45:47Z

Csheldon: /* Mobility for semiconductors */

2025-10-24T14:58:38Z

Mobility for semiconductors

← Older revision		Revision as of 14:58, 24 October 2025
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	\|}		\|}

	where <math>\sigma_{n \in \text{CB}}</math> and <math>\sigma_{n \in \text{VB}}</math> denote the conductivity restricted to states in the conduction and valence bands, respectively, <math>f_{n\mathbf{k}}</math> is the Fermi–Dirac distribution, and <math>\eta</math> is the [[Chemical potential in electron-phonon interactions \| chemical potential]] at the given temperature.		where <math>\sigma_{n \in \text{CB}}</math> and <math>\sigma_{n \in \text{VB}}</math> denote the conductivity restricted to states in the conduction and valence bands, respectively, <math>f_{n\mathbf{k}}</math> is the [[ELPH_FERMI_NEDOS \| Fermi–Dirac distribution]], and <math>\eta</math> is the [[Chemical potential in electron-phonon interactions \| chemical potential]] at the given temperature.
	{{NB\|mind\|To avoid confusion with the mobility, we use the notation <math>\eta</math>. Usually, the [[Chemical potential in electron-phonon interactions \| chemical potential]] is written as <math>\mu</math> or <math>\varepsilon_F</math>.\|}}		{{NB\|mind\|To avoid confusion with the mobility, we use the notation <math>\eta</math>. Usually, the [[Chemical potential in electron-phonon interactions \| chemical potential]] is written as <math>\mu</math> or <math>\varepsilon_F</math>.\|}}

Csheldon at 09:52, 24 October 2025

2025-10-24T09:52:54Z

← Older revision		Revision as of 09:52, 24 October 2025
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	and the Onsager coefficients are:		and the Onsager coefficients are:
	:<math>		:<math>
	L_{ij} = \int d\~~epsilon~~ \, \~~sigma~~(\varepsilon) \, (\varepsilon - \mu)^{i+j-2}		L_{ij} = \int d\varepsilon \, \mathcal{T}(\varepsilon) \, (\varepsilon - \eta)^{i+j-2}
	\left(-\frac{\partial f^0}{\partial \varepsilon}\right).		\left(-\frac{\partial f^0}{\partial \varepsilon}\right).
	</math>		</math>

Csheldon: /* Mobility for semiconductors */

2025-10-24T09:50:11Z

Mobility for semiconductors

← Older revision		Revision as of 09:50, 24 October 2025
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	The mobility is related to the conductivity $\sigma$, which can be defined using the transport distribution function.		The mobility is related to the conductivity $\sigma$, which can be defined using the transport distribution function.
	:<math>		:<math>
	\~~sigma~~(\varepsilon) = \frac{e^2}{N \Omega} \sum_{n\mathbf{k}} \tau_{n\mathbf{k}} \, \mathbf{v}_{n\mathbf{k}} \otimes \mathbf{v}_{n\mathbf{k}} \, \delta(\varepsilon_{n\mathbf{k}} - \varepsilon),		\mathcal{T}(\varepsilon) = \frac{e^2}{N \Omega} \sum_{n\mathbf{k}} \tau_{n\mathbf{k}} \, \mathbf{v}_{n\mathbf{k}} \otimes \mathbf{v}_{n\mathbf{k}} \, \delta(\varepsilon_{n\mathbf{k}} - \varepsilon),
	</math>		</math>
	and the Onsager coefficients are:		and the Onsager coefficients are:

Csheldon at 09:02, 24 October 2025

2025-10-24T09:02:05Z

← Older revision		Revision as of 09:02, 24 October 2025
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	\|}		\|}

	where <math>\sigma_{n \in \text{CB}}</math> and <math>\sigma_{n \in \text{VB}}</math> denote the conductivity restricted to states in the conduction and valence bands, respectively, <math>f_{n\mathbf{k}}</math> is the Fermi–Dirac distribution, and <math>\eta</math> is the chemical potential at the given temperature.		where <math>\sigma_{n \in \text{CB}}</math> and <math>\sigma_{n \in \text{VB}}</math> denote the conductivity restricted to states in the conduction and valence bands, respectively, <math>f_{n\mathbf{k}}</math> is the Fermi–Dirac distribution, and <math>\eta</math> is the [[Chemical potential in electron-phonon interactions \| chemical potential]] at the given temperature.
	{{NB\|mind\|To avoid confusion with the mobility, we use the notation <math>\eta</math>. Usually, the chemical potential is written as <math>\mu</math> or <math>\varepsilon_F</math>.\|}}		{{NB\|mind\|To avoid confusion with the mobility, we use the notation <math>\eta</math>. Usually, the [[Chemical potential in electron-phonon interactions \| chemical potential]] is written as <math>\mu</math> or <math>\varepsilon_F</math>.\|}}

	=== Constant relaxation-time approximation ===		=== Constant relaxation-time approximation ===

Csheldon at 12:49, 23 October 2025

2025-10-23T12:49:39Z

← Older revision		Revision as of 12:49, 23 October 2025
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	The most computationally demanding part of the calculation is the electronic linewidths due to the electron-phonon scattering.		The most computationally demanding part of the calculation is the electronic linewidths due to the electron-phonon scattering.
	As such, it is instructive to start by computing the transport coefficients in the constant-relaxation-time approximation (CRTA).		As such, it is instructive to start by computing the transport coefficients in the constant-relaxation-time approximation (CRTA).
	This gives us an idea of the <b>k</b>-point density required to obtain a precise value. See the [[Electronic transport coefficients \| theory page]] for more details on transport.		This gives us an idea of the <b>k</b>-point density required to obtain a precise value. See the [[Electronic transport coefficients \| theory page]] for more details on transport coefficients.
	{{Available\|6.5.0}}		{{Available\|6.5.0}}
	{{NB\|important\|This feature requires [[:Category:HDF5_support \| HDF5 support]].}}		{{NB\|important\|This feature requires [[:Category:HDF5_support \| HDF5 support]].}}

Csheldon at 07:13, 23 October 2025

2025-10-23T07:13:29Z

← Older revision		Revision as of 07:13, 23 October 2025
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	The transport coefficients can be evaluated rather straightforwardly under the approximation of the constant relaxation time.		The transport coefficients can be evaluated rather straightforwardly under the approximation of the constant relaxation time.
	The most computationally demanding part of the calculation is the electronic linewidths due to the electron-phonon scattering.		The most computationally demanding part of the calculation is the electronic linewidths due to the electron-phonon scattering.
	As such, it is instructive to start by computing the transport coefficients in the constant-relaxation-time approximation (CRTA).		As such, it is instructive to start by computing the transport coefficients in the constant-relaxation-time approximation (CRTA).
	This gives us an idea of the <b>k</b>-point density required to obtain a precise value.		This gives us an idea of the <b>k</b>-point density required to obtain a precise value. See the [[Electronic transport coefficients \| theory page]] for more details on transport.
	{{Available\|6.5.0}}		{{Available\|6.5.0}}
	{{NB\|important\|This feature requires [[:Category:HDF5_support \| HDF5 support]].}}		{{NB\|important\|This feature requires [[:Category:HDF5_support \| HDF5 support]].}}

Csheldon: /* Related tags and articles */

2025-10-21T10:38:31Z

Csheldon: /* Related tags and articles */

2025-10-20T14:23:03Z

← Older revision		Revision as of 11:10, 22 January 2026
Line 6:		Line 6:
	{{Available\|6.5.0}}		{{Available\|6.5.0}}
	{{NB\|important\|This feature requires [[:Category:HDF5_support \| HDF5 support]].}}		{{NB\|important\|This feature requires [[:Category:HDF5_support \| HDF5 support]].}}
			{{NB\|warning\|There was a [[Known_issues#KnownIssue54 \| known issue]] for electron-phonon calculations using {{TAG\|ISPIN\|2}} for VASP 6.5.0 and 6.5.1 that was fixed in VASP 6.6.0.}}

	The most basic {{FILE\|INCAR}} to compute the transport coefficients is		The most basic {{FILE\|INCAR}} to compute the transport coefficients is

← Older revision		Revision as of 10:38, 21 October 2025
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	* [[Electronic transport coefficients]]		* [[Electronic transport coefficients]]
	* [[Electron-phonon potential from supercells]]		* [[Electron-phonon potential from supercells]]
	* [[Chemical potential]]		* [[Chemical potential in electron-phonon interactions]]
	* [[Electron-phonon accumulators]]		* [[Electron-phonon accumulators]]
	* {{FILE\|phelel_params.hdf5}}		* {{FILE\|phelel_params.hdf5}}

← Older revision		Revision as of 14:23, 20 October 2025
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	== Related tags and articles ==		== Related tags and articles ==
	* [[Bandgap renormalization due to electron-phonon coupling\|Band-structure renormalization]]		* [[Bandgap renormalization due to electron-phonon coupling\|Band-structure renormalization]]
			* [[Electronic transport coefficients]]
	* [[Electron-phonon potential from supercells]]		* [[Electron-phonon potential from supercells]]
	* [[Chemical potential]]		* [[Chemical potential]]

← Older revision		Revision as of 09:45, 4 November 2025
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	::<math>\sigma = n_e \mu_e + n_h \mu_h</math>		::<math>\sigma = n_e \mu_e + n_h \mu_h</math>
	with <math>n_e</math> being the density of electron carriers, <math>n_h</math> the density of hole carriers and <math>\mu_e = \sigma_e/n_e</math> and <math>\mu_h = \sigma_h/n_h</math> with <math>\sigma_e</math> being the contribution of the conduction bands to the electron conductivity and <math>\sigma_h</math> of the valence bands.		with <math>n_e</math> being the density of electron carriers, <math>n_h</math> the density of hole carriers and <math>\mu_e = \sigma_e/n_e</math> and <math>\mu_h = \sigma_h/n_h</math> with <math>\sigma_e</math> being the contribution of the conduction bands to the electron conductivity and <math>\sigma_h</math> of the valence bands.
			{{NB\|important\|Be careful not to mistake the [[Chemical potential in electron-phonon interactions \| chemical potential]] <math>\mu</math> for the mobility <math>\mu_e</math> or <math>\mu_h</math>.}}
	For the computation of a realistic mobility, it is usual to compute the conductivity for different doping levels and extrapolate to the limit of the lowest possible carrier concentration. The value of the mobility should be relatively constant in the limit of low carrier concentrations.		For the computation of a realistic mobility, it is usual to compute the conductivity for different doping levels and extrapolate to the limit of the lowest possible carrier concentration. The value of the mobility should be relatively constant in the limit of low carrier concentrations.
	Additionally, one can specify electron or hole doping. A convenient setting is		Additionally, one can specify electron or hole doping. A convenient setting is
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	and the Onsager coefficients are:		and the Onsager coefficients are:
	:<math>		:<math>
	L_{ij} = \int d\varepsilon \, \mathcal{T}(\varepsilon) \, (\varepsilon - \~~eta~~)^{i+j-2}		L_{ij} = \int d\varepsilon \, \mathcal{T}(\varepsilon) \, (\varepsilon - \mu)^{i+j-2}
	\left(-\frac{\partial f^0}{\partial \varepsilon}\right).		\left(-\frac{\partial f^0}{\partial \varepsilon}\right).
	</math>		</math>
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	\| Electron mobility <math>\mu_e</math>		\| Electron mobility <math>\mu_e</math>
	\| <math>\mu_e = \tfrac{\sigma_{n \in \text{CB}}}{n_e}</math>		\| <math>\mu_e = \tfrac{\sigma_{n \in \text{CB}}}{n_e}</math>
	\| <math>n_e = \frac{1}{\Omega N_\mathbf{k}}\sum_{\mathbf{k}n \in \text{CB}} f(\varepsilon_{\mathbf{k}n}, T, \~~eta~~)</math>		\| <math>n_e = \frac{1}{\Omega N_\mathbf{k}}\sum_{\mathbf{k}n \in \text{CB}} f(\varepsilon_{\mathbf{k}n}, T, \mu)</math>
	\|-		\|-
	\| Hole mobility <math>\mu_h</math>		\| Hole mobility <math>\mu_h</math>
	\| <math>\mu_h = \tfrac{\sigma_{n \in \text{VB}}}{n_h}</math>		\| <math>\mu_h = \tfrac{\sigma_{n \in \text{VB}}}{n_h}</math>
	\| <math>n_h = \frac{1}{\Omega N_\mathbf{k}}\sum_{\mathbf{k}n \in \text{VB}} \big[1 - f(\varepsilon_{\mathbf{k}n}, T, \~~eta~~)\big]</math>		\| <math>n_h = \frac{1}{\Omega N_\mathbf{k}}\sum_{\mathbf{k}n \in \text{VB}} \big[1 - f(\varepsilon_{\mathbf{k}n}, T, \mu)\big]</math>
	\|}		\|}

	where <math>\sigma_{n \in \text{CB}}</math> and <math>\sigma_{n \in \text{VB}}</math> denote the conductivity restricted to states in the conduction and valence bands, respectively, <math>f_{n\mathbf{k}}</math> is the [[ELPH_FERMI_NEDOS \| Fermi–Dirac distribution]], and <math>\~~eta~~</math> is the [[Chemical potential in electron-phonon interactions \| chemical potential]] at the given temperature.		where <math>\sigma_{n \in \text{CB}}</math> and <math>\sigma_{n \in \text{VB}}</math> denote the conductivity restricted to states in the conduction and valence bands, respectively, <math>f_{n\mathbf{k}}</math> is the [[ELPH_FERMI_NEDOS \| Fermi–Dirac distribution]], and <math>\mu</math> is the [[Chemical potential in electron-phonon interactions \| chemical potential]] at the given temperature.
	{{NB\|mind\|To avoid confusion with the mobility, we use the notation <math>\eta</math>. Usually, the [[Chemical potential in electron-phonon interactions \| chemical potential]] is written as <math>\mu</math> or <math>\varepsilon_F</math>.\|}}

	=== Constant relaxation-time approximation ===		=== Constant relaxation-time approximation ===